Analyse du transfert de chaleur par ébullition et condensation à l’Intérieur d’un caloduc horizontal

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Faculté de génie

Département de génie chimique et de génie biotechnologique

Analyse du Transfert de Chaleur par Ébullition et

Condensation à l’Intérieur d’un Caloduc Horizontal

Thèse de doctorat

Spécialité : génie chimique

Roshanak Rabiee

Jurés:

Prof. Martin Désilets (Directeur)

Prof. Pierre Proulx (co-directeur)

Prof. Jocelyn Veilleux (Rapporteur)

Prof. Louis Gosselin (Examinateur)

Prof. Sébastien Poncet (Examinateur)

Université de Sherbrooke

Sherbrooke (Québec) Canada

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Document adopté à la Faculté de génie de l’Université de Sherbrooke par le Comité de la recherche et des études supérieures et le Comité des programmes (2019).

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L'utilisation de différents types de caloducs dans les systèmes de climatisation, de ventilation et d'évacuation de la chaleur a considérablement augmenté. Les caloducs sont des dispositifs de transfert de chaleur à deux phases générant un flux de chaleur élevé avec un faible gradient de température et une perte de charge minimale. La réduction des coûts de maintenance est l'un des avantages de l'utilisation de caloducs, due à l'absence de pièces mécaniques, à la réduction de l'espace occupé et à la surveillance, la fabrication et la maintenance simplifiées. Les caloducs sont des conducteurs thermiques très efficaces en raison des flux de chaleur élevés obtenus lors de l’évaporation et de la condensation du fluide de travail. Pour optimiser les performances d'un caloduc, il est nécessaire d'étudier précisément ce qui se passe à l'intérieur des sections de l'évaporateur et du condenseur.

Ce projet consiste à développer un modèle numérique simulant un écoulement diphasique à l'intérieur d'un caloduc à l'aide de codes CFD développés dans OpenFOAM. Le modèle est capable de prédire les principales variables représentant le comportement des deux phases telles que la vitesse, la température, la pression et la fraction volumique de chaque phase dans l’évaporateur ou le condenseur. Une attention particulière est consacrée à la simulation de la condensation car il n'existe pas de modèle numérique de ce type dans la littérature pour l'analyse du transfert de chaleur par condensation. En outre, une combinaison d'ébullition et de condensation dans un caloduc est une autre contribution de ce travail. Dans ce projet, le modèle de fractionnement du flux thermique de la paroi dans le cadre de l’approche eulérienne en deux phases a été appliqué. L'effet du transfert de quantité de mouvement et du transfert d'énergie entre les deux phases est également pris en compte.

La capacité du modèle numérique a été validée par des données expérimentales obtenues à partir d’essais réalisés sur un prototype construit à l’Université de Sherbrooke. Ensuite, un modèle validé est utilisé pour évaluer les performances de ce caloduc.Enfin, deux types de structures de gorge ont été suggérés et des tests expérimentaux ont été effectués pour étudier toute amélioration des performances du caloduc.

Mots-clés : flux en deux phases, modèle d'Euler-Euler, caloduc, ébullition et condensation,

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The usage of different types of heat pipe in air conditioning, ventilation and heat removal systems has tremendously increased. Heat pipes are two-phase heat transfer devices and they are generating high heat flux with minimum temperature gradient and pressure drop. Lower service costs, due to the absence of mechanical parts, less occupied space and easier monitoring, manufacturing and maintenance are some of the advantages of using heat pipes. Heat pipes are highly effective thermal conductors due to the high heat fluxes obtained during boiling and condensation. For the optimization of heat pipe performances, it is necessary to study what happens inside the evaporator and condenser sections, precisely.

This project deals with the development of a numerical model to simulate the two-phase flow inside of the heat pipe using CFD codes developed in OpenFOAM. The model is able to predict the main two-phase variables such as velocity, temperature, pressure and volume fractions of each phase in the evaporator or condenser. Special attention is devoted to the simulation of condensation since there is no such numerical model in the open literature for the analysis of condensation heat transfer. In addition, the combination of boiling and condensation inside the heat pipe is another contribution of this work. In this project, a wall heat flux partition model in the framework of two-phase Eulerian approach has been applied. The effect of the interfacial momentum and energy transfer between the two phases is also taken account.

The capability of numerical model was validated by comparing numerical prediction with experimental data obtained from tests which have been conducted using a built-up prototype designed at Université de Sherbrooke. Then, the validated numerical model is used to assess the performances of this heat pipe. Finally, two types of grooves have been suggested and some experimental tests were performed to investigate any improvement in the heat pipe performance.

KeywordsTwo-phase flow, Euler-Euler model, heat pipe, boiling and condensation, thermal resistance, horizontal smooth tube, area fraction.

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This thesis is dedicated to my mother and my family who always loved and supported me. A special acknowledgement should be made to my mother for her patience and love along the completion of this degree. I would like to acknowledge my beloved father and dedicate this thesis to the memories he hacked in my heart.

I would like to thank my supervisors, Prof. Désilets, for guiding me and helping me to reach my potential. Prof. Désilets is admirable for both personal and professional attributes that he has. Many thanks to Prof. Proulx for his helps to complete this degree. I learned a lot from his collaboration.

I would also like to warmly thank my committee members, Professor Veilleux, Professor Poncet and Professor Gosselin for their precise and valuable comments.

Also, want to thank all my colleagues at UdeS, who have been good friends. Being a part of this group was a pleasure for me.

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1. CHAPTER ONE: Introduction ... 1

1.1 Heat exchanger and application of heat pipe ... 1

1.2 Operation principles of heat pipe... 3

1.3 A brief review of classification of heat pipes ... 4

1.4 Operating limits of a heat pipe ... 7

1.5 Heat pipe function analysis ... 9

1.6 Heat pipe performance ... 12

1.7 Research project description ... 13

1.8 Objectives ... 15

1.9 Study plan ... 15

1.10 Methodology ... 16

2 CHAPTER TWO: State of the art ... 20

2.1 Phase change concepts ... 20

2.1.1 Boiling concepts ... 20

2.1.2 Condensation concepts ... 25

2.2 Numerical, experimental and analytical modeling of heat pipes ... 28

2.3 Modeling phase change ... 31

2.3.1 Advantages of Two-Fluid model ... 33

2.3.2 Governing equations of Two-Fluid Model ... 34

3 CHAPTER THREE: Boiling Model ... 38

3.1 Literature review ... 38

3.2 Wall heat flux ... 39

4 CHAPTER FOUR: Determination of Heat Transfer for Flow Condensation inside of Horizontal Smooth Tube ... 46

4.1 Abstract ... 46

4.2 Literature review ... 46

4.3 Governing equations ... 49

4.4 Boiling Wall heat flux ... 50

4.5 Condensation wall heat flux ... 52

4.5.1 Proposed Correlation for Heat Transfer Coefficient in Condensation ... 53

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4.7 Numerical Solution ... 57

4.7.1 Results and Discussion ... 59

4.7.2 Validation of Simulation Results ... 60

4.7.3 Effect of Mass Flux, Diameter and Sub-Cooled Temperature ... 65

4.7.4 Condensation Number ... 68

4.8 Conclusion ... 70

5 CHAPTER FIVE: Heat Transfer Analysis of Boiling and Condensation in Horizontal Heat Pipe ... 74

5.1 Abstract ... 74

5.2 Literature review ... 74

5.3 Experimental works ... 78

5.3.1 Position of thermistors and accuracy of measurement data ... 79

5.3.2 Test procedure ... 81

5.3.3 Results of experimental works ... 82

5.4 Mathematical formulation ... 86

5.4.1 Governing equation ... 86

5.4.2 Boiling Model ... 86

5.4.3 Condensation Model ... 88

5.5 Validation of Numerical model ... 90

5.5.1 Boundary conditions and numerical model ... 91

5.6 Results and Discussion ... 93

5.6.1 Temperature distribution analysis ... 95

5.6.2 Heat pipe performance analysis ... 95

5.7 Parametric analysis, optimization of model ... 97

5.7.1 Effect of heat input on heat pipe performance... 97

5.7.2 Effect of filling ratio on heat pipe performance ... 98

5.7.3 Effect of diameter of channel on heat pipe performance ... 100

5.7.4 Effect of the type of refrigerant on the heat pipe performance ... 101

5.8 Conclusion ... 102

6 CHAPTER SIX: Effect of groove on heat pipe performance ... 103

6.1 Abstract ... 103

6.2 Literature review ... 103

6.3 Experimental works ... 106

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6.4 Comparing heat pipe performance ... 110

6.5 Conclusion ... 113

7 CHAPTER SEVEN: Conclusion and futures works ... 114

8 References ... 120

9 Appendix ... 125

9.1 Capillary limit ... 125

9.2 Euler-Euler or Two-Fluid model ... 126

9.2.1 Modeling lift force ... 126

9.2.2 Modeling drag force ... 127

9.2.3 Modeling virtual mass force ... 129

9.2.4 Modeling wall lubrication force ... 129

9.2.5 Modeling turbulent dispersion force ... 130

9.2.6 Two-phase turbulent modeling using k-ε ... 130

9.3 Mass and Heat flux at boundary and bulk fluid ... 131

9.3.1 Interphase heat transfer (boiling or condensation) inside of bulk fluid ... 132

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Figure 1-1: Heat pipe as part of a ventilation system-adapted figure from [2] ... 2

Figure 1-2: Vapor-liquid interface in a closed heat pipe-adapted figure from- [1]... 4

Figure 1-3: Gravity-assisted wickless heat pipe, thermosyphon- adapted figure from- [1] ... 5

Figure 1-4: Conventional horizontal heat pipe with wick structure- adapted figure from- [1] ... 5

Figure 1-5: A sample of rotating heat pipe- adapted figure from- [1] ... 6

Figure 1-6: a) Un-looped pulsating heat pipe, b) looped pulsating heat pipe- adapted figure from- [3] ... 7

Figure 1-7: Heat transfer limitation in a heat pipe - adapted figure from [6] ... 9

Figure 1-8: Flow chart of heat pipe operation and interaction between different regions- adapted figure from- [1] ...10

Figure 1-9: a) Various components of heat pipe- b) Temperature-Entropy diagram in thermodynamic cycle of heat pipe - adapted figure from- [1] ...11

Figure 1-10: Thermal resistance across a horizontal heat pipe- adapted figure from [6] ...11

Figure 1-11: Prototype of heat pipe build in chemical engineering laboratory in UdeS ...14

Figure 1-12: Structure of code in OpenFOAM- [13] ...18

Figure 1-13: Simulation of flow condensation in 3-D cylindrical channel ...18

Figure 1-14: Complete model of U-type smooth heat pipe ...19

Figure 2-1: Boiling curve, heat flux vs superheat temperature- adapted figure from [15], [9] ...21

Figure 2-2: flow boiling regime from stratified to annular regime [9] ...23

Figure 2-3: pool condensation regime dropwise and filmwise- adapted figure from [29]...26

Figure 2-4: flow condensation regime [29] ...27

Figure 2-5: (a) Surface-tracking method, (b) moving mesh method (c) volume-tracking method -[58] ...32

Figure 2-6: Averaged modelling approaches for two-phase flow → (Euler-Lagrange) model-[58] ...32

Figure 2-7: Averaged modeling approaches for two-phase flow→Two-Fluid (Euler-Euler) model-[58] ...33

Figure 3-1: Wall boiling model consisting of evaporation, quenching and single-phase convection ...40

Figure 3-2: waiting time and contact time for departed bubble when detaches at nucleation site-[64] ...42

Figure 3-3: algorithm of wall heat flux portioning model in OpenFOAM ...43

Figure 4-1: Wall boiling model consisting of evaporation, quenching and single-phase convection ...51

Figure 4-2: Wall condensation model consists of condensation heat flux and single-phase convection ...53

Figure 4-3: Geometry and boundary condition of tube ...56

Figure 4-4: 3D-hexahedral mesh and gravity direction ...56

Figure 4-5. Simulation results into flow pattern map of Suliman -[46] ...59

Figure 4-6: Deviation graph for total heat transfer coefficients of case I & case II...62

Figure 4-7: Local heat transfer coefficients versus experimental results for different mass fluxes-case I ....63

Figure 4-8: Local heat transfer coefficients versus experimental results for different mass fluxes-case II ...63

Figure 4-9: (a), (b)- Local heat transfer coefficients versus vapor quality- at three mass fluxes G=125, 200 kg/m2.s for case I & Fig. 9 (c), (d) G=300, 400 (kg/m2.s) for case II, compared with experimental result and correlations data ...65

Figure 4-10: Local heat transfer coefficient versus vapor quality at different mass fluxes, D=9 mm, L=1 m, dTsub=3°C, Tsat=40 oC ...66

Figure 4-11: Flow pattern map to identify the temperature dependency ...67

Figure 4-12: Relation between the condensation number and the liquid Reynolds number for different mass fluxes ...69

Figure 4-13: Comparison between total heat transfers calculated by new suggested correlation and the experimental data ...70

Figure 5-1: Schematic view of experimental test setup ...78

Figure 5-2: View of build-up experimental setup ...79

Figure 5-3: Position of thermistors in heat pipe ...80

Figure 5-4: Temperature history of wall power 24W- case I ...84

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Figure 5-9: 3D-hexahedral mesh and gravity direction ...90

Figure 5-10: Polynomial relation between saturation pressure and temperature for R134a ...93

Figure 5-11: Temperature distribution versus heat input, a comparison between numerical model and thermistors measurements ...95

Figure 5-12: Evaporator and condenser thermal resistances: experimental versus numerical values ...97

Figure 5-13: Effect of heat input on equivalent thermal resistance ...98

Figure 5-14: Effect of the filling ratio on equivalent thermal resistance ...99

Figure 5-15: Effect of the diameter of tube on the equivalent thermal resistance for filling ratio 50% at different powers ...100

Figure 5-16: Effect of the type of refrigerants on the equivalent thermal resistance ...101

Figure 6-1: Different types of commercial enhanced tube for pool boiling- [85], [86] ...103

Figure 6-2: Different types of 3-dimensional tubes for condensation-[86] ...104

Figure 6-3: (a) Non-anodized surface, (b) anodized surface [79]-(c) surface with pure working fluid water (d) surface after adding nanoparticle to working fluid (40 nm Cu)- [90] ...105

Figure 6-4: Position of thermistors on groove heat pipe...107

Figure 6-5:Non-dimensional width and space between groove vs height ...107

Figure 6-6: Schematic of smooth, helicoidal and axial grooved pipe ...108

Figure 6-7: Wall temperature history of helicoidal grooved pipe at 24 W power ...109

Figure 6-8: Wall temperature vs time for axial grooved pipe at 24W power ...110

Figure 6-9: Equivalent thermal resistance for three different designs ...111

Figure 6-10: Evaporator thermal resistance for three different design ...112

Figure 6-11: Helicoidal grooved pipe in saturated boiling region at power 110 W ...113

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TABLE 1-1: Thermal resistance model in heat pipe and their order of magnitude [6] ... 12

TABLE 1-2: Some available solvers in OpenFOAM to model phase changes ... 17

TABLE 2-1: Proposed correlation for pool boiling regime ... 22

TABLE 2-2: Proposed correlation for flow boiling regime ... 23

TABLE 2-3: Proposed correlations for film condensation and dropwise regime ... 26

TABLE 2-4: Proposed correlation for flow condensation regime... 28

TABLE 2-5: Reference works for modeling heat pipes ... 29

TABLE 4-1: reference correlation used for calculation of condensation area fraction ... 54

TABLE 4-2: Grid sensitivity analysis ... 57

TABLE 4-3: Properties and geometry of Ref [46]–case I ... 58

TABLE 4-4: Properties and geometry of Ref [45]–case II ... 58

TABLE 4-5: Total heat transfer coefficient for case I ... 61

TABLE 4-6: Total heat transfer coefficient for case II ... 61

TABLE 4-7: effect of sub-cooled temperature on heat transfer coefficient ... 68

TABLE 4-8: effect of hydraulic dimeter on heat transfer coefficient ... 68

TABLE 4-9: Validation of suggested correlation for condensation heat transfer ... 70

TABLE 5-1: Uncertainty of measurement devices ... 81

TABLE 5-2: Properties of refrigerant at saturation point and properties of water at cooling conditions ... 83

TABLE 5-3: Grid sensitivity analysis ... 91

TABLE 5-4: Properties and geometry of numerical simulation ... 92

TABLE 5-5: Comparison between numerical estimation with experimental ... 94

TABLE 6-1: Specification of tests tubes ... 108

TABLE 9-1: Minimum effective radii of curvature and wetting angles for typical fluids ... 125

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List of Symbols

Latin Symbols

A Surface area m2

w

A

Ab Boiling area fraction ---

Ac Condensation area fraction ---

aL Experimental parameter ---

CD Drag coefficient ---

CL Lift coefficient ---

CwL Wall lubrication coefficient ---

CP Specific heat capacity kJ/kg.K

Csf Experimental constant that depends on surface-fluid

combination ---

Chl Liquid Stanton number ---

Cf Fanning friction factor ---

Co Condensation number ---

Csf Experimental constant for surface-fluid interaction ---

w

d

c w

d Liquid droplet diameter m

D Hydraulic diameter m

f Detachment frequency Hz or s-1

g Gravity m/s2

h Heat transfer coefficient W/m2_.K

i Specific Enthalpy J/kg

It _{Identity matrix ---}

I Electrical current A

Jv Dimensionless vapor velocity ---

G Inlet mass flux kg/m2_.s

hlv Latent heat (vaporization latent heat) J/kg

Hc Stanton heat transfer coefficient ---

kt _{Kinetic energy of turbulent J/kg}

L Length of tube m

qa Conductive heat flux W/m2

t a

q Turbulent conductive heat flux W/m2

" wall

q Wall heat flux W/m2

c" wall

q Single-phase convective heat flux W/m2

e" wall

q Evaporation heat flux W/m2

q" wall

xi

𝑀̅ Molecular weight kg/mole

Ma Interfacial momentum exchange between two phases N.s

e

m

cd

m

𝑁" Nucleation site density m-2

𝑅̅ Universal gas constant kJ/mol.K

Nu Nusselt number --- Pr Prandtl number --- P Pressure Pa Po Power W Pr Reduced pressure --- U Velocity m/s

𝑈𝑟 Relative velocity between two phases m/s

V Voltage drop V

𝑀̅_𝑎 Averaged inter-phase momentum transfer N.s

𝑀_𝑎𝑑 _{Momentum of drag force N.s}

𝑀_𝑎𝑙 Momentum of lift force N.s

𝑀_𝑎𝑤𝑙 _{Momentum of lubrication force N.s}

𝑀_𝑎𝑣𝑚 _{Momentum of virtual mass force N.s}

𝑀𝑎𝑡𝑑 Momentum of turbulent dispersion force N.s

Re Reynolds number ---

R Thermal resistance K/W

Rpe Radial thermal resistance of evaporator K/W

Rpc Radial thermal resistance of condenser K/W

Rint.evap Thermal resistance of boiling K/W

Rint.cond Thermal resistance of condenser K/W

Rpa Axial thermal resistance of wall K/W

Rwa Axial thermal resistance of wick K/W

T Temperature K,o_C

Tr Reduced temperature ---

t Time s

x Vapor quality (see Eq.4.24) ---

Xtt Martinelli parameter --- Greek symbols 𝜌 Density kg/m3 𝛼 Phase fraction --- ba +

 Evaporation rate per unit volume kg/s.m3

ab +

 Condensation rate per unit volume kg/s.m3

𝜇 Dynamic viscosity kg/s.m3

xii w



λ Thermal conductivity W/m.K

 Laminar viscosity stress N/m2

t

 Turbulent viscosity stress N/m2

𝜎 Surface tension kg/s2 P











∆ Delta or difference operator ---

∇ Gradient operator ---

subscripts

adj _{Adjacent to the wall}

a _{Phase a or vapor} b _{Phase b or liquid} crt _Critical cap _Capillary cond _Condenser corr _Correlation evap _Evaporator equiv _Equivalent exp _Experimental fric _Friction int _Interfacial l _Liquid num _Numerical ref _Reference sat _Saturation sim _Simulation static _Static sub _Subcooled sup _Superheat v _Vapor wall _Wall w _Water t _Turbulent

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1. CHAPTER ONE: Introduction

1.1 Heat exchanger and application of heat pipe

The common challenge in heating and cooling industry is to recover as much thermal energy as possible from waste air in order to heat incoming fresh air and save substantial energy during air conditioning or heating of buildings. Also, conventional heating and cooling systems occupy more space and they usually require fire control system, smoke damper and also pressure loss in duct which leads to higher energy consumption on air transport. However, one main objective of designing a heating and cooling system is to increase its thermal efficiency, occupy less space, having high heat transfer rate without pressure drop across the exchanger. High-pressure drop also causes a lot of noises in heat exchanger and also increases the manufacturing, operations and maintenance costs. Heat pipes are among new air conditioning technologies that can substitute conventional systems. Their lower service costs, higher thermal efficiency and easier monitoring, applicability, manufacturing and maintenance are some of the advantages of using heat pipes explaining increasing demand for using them in the design of heat and ventilation systems. Due to the high heat transfer coefficients in boiling and condensation regime, heat pipes are highly effective thermal conductors. The effective thermal conductivity varies with heat pipe length, and can approach 100 kW/m⋅K for long heat pipes, in comparison with approximately 0.4 kW/m⋅K for copper with the same dimension. A 15-cm-long, 0.6-cm-diameter horizontal cylindrical heat pipe with water inside, for example, can transfer 300W of heat with less than a 10 °C temperature drop from one end to the other. Some heat pipes have demonstrated a heat flux of more than 23 kW/cm². Because of these reasons, they are widely used in different areas such as heat sinks for cooling electrical components and computers, spacecraft thermal control system, solar thermal, permafrost cooling, ventilation or recovery systems and heat exchangers [1].

Heat pipe generally consists of an evaporator, condenser, adiabatic section and a working fluid, which undergoes a phase change process from liquid to vapor (boiling) in the evaporator and vapor to liquid (condensation) in the condenser. In other words, the working fluid rapidly absorbs large amounts of energy as latent heat of vaporization from a heat source and releases

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it to the cooler side without any mechanical parts. Adiabatic section is only an intermediate part that connects the two sections.

Figure 1-1: Heat pipe as part of a ventilation system-adapted figure from [2]

In order to analyze a heat pipe model, it is necessary to simulate each section and the phase changes that happened, precisely. The challenging work in the simulation of evaporator is the modeling of bubble formation that strictly affects the heat transfer rate. Moreover, there are few numerical works for estimation of heat transfer in the condensation regimes. The combination of boiling and condensation inside a heat exchanger like a heat pipe definitely brings challenging difficulties to solve. This complexity is increased even more when the system is a closed one because of small pressure variations in such applications.

The critical items for the design of an optimum heat pipe are the pipe material, dimensions, type of working fluid, operating conditions and surface roughness. The operation of a heat pipe is based on phase change and liquid-vapor interfaces. If the working fluid is heated above a certain temperature, evaporator would dry out and in this case, the thermal efficiency of the heat pipe is suddenly reduced. On the other hand, below a certain temperature, the working fluid will not undergo boiling and the evaporator will not work properly, the heat pipe will be flooded by condensed vapor and again its performance is considerably degraded. The pipe dimensions

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control the surface area and consequently influence the heat transfer rate. The type of working fluid and pipe material is influencing the liquid-vapor interface and affect phase change processes, directly. Surface roughness like existing grooves will increase the density of nucleation sites for initiating boiling and condensation. Therefore, it is necessary to evaluate the effect of these parameters on heat transfer for optimizing the behavior of a heat pipe.

1.2

Operation principles of heat pipe

Generally, a heat pipe consists of three main sections: an evaporator section, a condenser section and an adiabatic section (Figure 1-2). These components are inside of a sealed container (pipe wall and end caps) and an amount of working fluid circulates between sections. In some heat pipes, a wick structure, grooves or coating are added to the wall for helping the circulation of flow between the evaporator and condenser. However, based on the type of heat pipes, other forces like gravity, capillary, centrifugal, electrostatic and osmotic forces can help flow circulation between the two main sections. The working fluid should have the following properties: high latent heat of vaporization, high thermal conductivity, low viscosity and high surface tension.

When the heat is externally applied to the evaporator section, the saturated liquid vaporizes, and the pressure of the evaporator increases due to accumulation of vapor at the top. This phenomenon is pushing vapor flow towards the condenser where it condenses to liquid at the contact of the cooled walls maintained at a temperature below the saturation point. The liquid film at top section of the condenser falls down to the bottom where it accumulates and returns to the evaporator section. The vapor and liquid circulate continuously between evaporator and condenser due to pressure differences. Therefore, the heat pipe can continuously transport the latent heat of vaporization from the evaporator (hot zone) to the condenser section (colder zone) [3].

4 liquid

vapor

Heat Source Heat Sink

evaporator condenser

adiabatic section

Figure 1-2: Vapor-liquid interface in a closed heat pipe-adapted figure from- [1]

1.3 A brief review of classification of heat pipes

There are various types of heat pipes used in air conditioning system including vertical, horizontal, annular, rotating and pulsating heat pipes. In this section, the main types of heat pipes are briefly introduced:

• Two-Phase vertical heat pipe or thermosyphon: In the case of a thermosyphon, the evaporator is located below the condenser section. There is no wick structure since gravity force can return liquid to evaporator. When heat is applied to the evaporator, the liquid is vaporized and flows upward to the condenser section. Vapor is then condensed on cooled walls and returns back to the evaporator section along the wall with the help of the gravity force (Figure 1-3).

5 vapor cond ens er eva pora tor adi aba ti c se ct ion liquid gravity

Figure 1-3: Gravity-assisted wickless heat pipe, thermosyphon- adapted figure from- [1]

• Horizontal, capillary heat pipe: In these heat exchangers, beside the evaporator, condenser and adiabatic sections, a wick structure is usually added to assist the liquid in returning to the evaporator. When heat is applied, the working fluid absorbs the latent heat and vaporizes. Then, the vapor moves to the condenser section driven by the pressure difference where it releases its latent heat to an external cooling system. Liquid then returns to the evaporator helped by capillary forces. (Figure 1-4).

vapor flow

liquid flow

evaporator condenser

adiabatic section

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• Rotating heat pipes: In this type, a centrifugal force drives back the liquid to the evaporator (Figure 1-5). In disk-shaped heat pipes, heat is provided at the outer radius and extracted at the inner radius, which allows the flow to be driven back to the evaporator by the centrifugal force with the aid of an internal taper. Therefore, a wick structure is not used in rotating heat pipes. The cylindrical and disk rotating heat pipe are used in electrical motors, metal cutting tools, turbine components and automobile brakes.

condenser evaporator Axis of rotation evaporator vapor flow liquid flow

Figure 1-5: A sample of rotating heat pipe- adapted figure from- [1]

• Pulsating heat pipes: A long tube is bent into many sections, which causes liquid slug and vapor plug to fluctuate along the channel, periodically. The diameter of pulsating heat pipe is usually small (less than 5 mm) and vapor plug and liquid slug are formed across pipe as a result of capillary effect. An oscillating movement of vapor plug and liquid slug is generated due to pressure difference between cooling section and heating section. A sample of Un-looped and looped pulsating heat pipes is shown in Figure 1-6.

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Figure 1-6: a) Un-looped pulsating heat pipe, b) looped pulsating heat pipe- adapted figure from- [3]

1.4 Operating limits of a heat pipe

The limiting parameters in the heat transport of conventional heat pipes are: viscous limit, sonic limit (shock happened at the end of the evaporator like at the throat of a nozzle), entrainment limit, capillary limit, boiling limit [3].

• Viscous limit: it mostly happens during start up, when the operating temperature is extremely low and the applied heat is small. In these conditions, the viscous force is larger than the pressure gradient and the liquid stream cannot overcome the friction force to circulate between sections.

• Sonic limit: with the increasing velocity of vapor in a long heat pipe, it can reach to sonic speed and a chock happened at the end of evaporator section. The sonic limit usually occurs in startup or steady condition of very long heat pipe.

• Entrainment limit: at high velocity, the shear forces at the liquid-vapor interface are increased and push the liquid to the condenser section. If the entrainment is too high, the evaporator can dry out. The entrainment limit has been seen in low temperature heat pipes with small diameters and high temperature heat pipe when applied heat at evaporator section is high.

• Capillary limit: this limitation is more critical in the design of low temperature heat pipes when the capillary pressure is not sufficient to return liquid to the evaporator section. Any attempt to increase the heat transfer above the capillary limit causes dry out of the evaporation section [3]. Capillary pressure must be greater than total pressure loss in order to drive liquid back to the evaporator. The total pressure drop in the system



P

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sum of the pressure drop of vapor and liquid due to friction, hydrostatic pressure difference, pressure drops of vapor in the evaporator and the condenser [4].

Ptotal =  +  = Pl Pv Pl,fric+ Pl static, + Pv evap, + Pl cond, (1.1)

Based on Young-Laplace equation, the capillary force must develop a pressure greater than the total pressure drop, under normal operation [more explanations are found in Appendix section 9.1]

,max

cap total

P P

   (1.2)

• Boiling limit: It happens due to high value of applied heat at evaporator, a situation which leads to dry out. The vapor bubbles that form in wick structure prevent the liquid to reach and wet the wall surface. The boiling limit is usually seen in a heat pipe with non-metallic working fluid [5].

Launay et al. [4] concluded that the boiling and capillary limit are the two main limitations in heat pipe performance and other factors have negligible effects. In order to increase the capillary force and boiling limit, the pore size in the evaporator, which acts as a nucleation site, should be as low as possible and the saturation pressure must be high enough. As shown in Figure 1-7, the maximum heat transport capability of heat pipe is constrained by the five limits mentioned before. A steady-state operating mode can only be reached in the area below this curve [6].

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Figure 1-7: Heat transfer limitation in a heat pipe - adapted figure from [6]

1.5 Heat pipe function analysis

The thermal-fluid phenomena occurring in a heat pipe can be divided into four basic categories: (1) heat conduction in the wall or wick; (2) liquid flow in the wick structure; (3) interaction between the liquid and vapor and (4) vapor flow in the core region. The interaction between these regions is illustrated in (Figure 1-8).

10 Heat source-input Heat sink-output Vapor space Wick Structure Wall Wick Structure Wall 1 2 3 4 Temperature Heat Input Heat Output Conjugate Effects Radiation or Convection B.C.s Geometry Effect Heat Conduction Stefen-Boltzman Law Fourier Law

Variables Phenomena Governing Equations

(1) Heat Conduction in Heat Pipe Container

Temperature Velocities Pressure drop

Gravity

Phase change Porous media flow Interfacial Shear Stress

Continuity equation Modified Navier-Stokes Equation

Averaged Energy Equation

Variables Phenomena Governing Equations

(2) Liquid flow & Heat Transfer in Wick structure

Contact angle Interfacial curvature Interfacial Positions Interfacial Mass Flux

Capillary Pressure Evaporation Condensation

Interfacial Mass Balance Interfacial Momentum Balance

Interfacial Energy Balance Clapeyron Equation

Variables Phenomena Governing Equations

(3) Liquid-Vapor Interface Velocities Pressure drop Density Temperature Compressibility Effects Mass Diffusion Geometry Effect Sonic Limitation Continuity equation Momentum equation Energy equation Mass Diffusion equation

Equation of state

Variables Phenomena Governing Equations

(4) Vapor flow in Heat pipe core

Heat Source Heat Sink

Figure 1-8: Flow chart of heat pipe operation and interaction between different regions- adapted figure from- [1] Heat pipe components and idealized thermodynamic cycle of operation are depicted in Figure 1-9-a. A quantity of heat Qin is applied to evaporator section. Under steady state, the same

quantity of heat is rejected at the condenser. The work, enclosed surface in T-S diagram, is generated inside of heat pipe and used to compensate losses in system and draw liquid back to the evaporator section.

11 P2 P3 Evap orator _Te Condenser Tc

Vapor flow from evaporator to condenser

liquid flow from condenser to evaporator Qin Qout 2 1 3 4 Qin Qout

Figure 1-9: a) Various components of heat pipe- b) Temperature-Entropy diagram in thermodynamic cycle of heat pipe - adapted figure from- [1]

According to Figure 1-9-b, the fluid enters the evaporator as a compressed liquid at T1 and receives heat. Then, it leaves evaporator at point 2 or 2’ as superheated or saturated vapor. Due to the pressure difference between the evaporator and the condenser (P2>P3), the vapor flows to

condenser section in path 2-3 or 2’-3. Then, the vapor enters the condenser as saturated vapor and loses its latent heat. Next, it enters adiabatic section as saturated liquid (point 4). Finally, the liquid leaves the adiabatic section to enter evaporator as a compressed liquid and completes cycle (point 1). If one considers the heat pipe as thermal circuit, a different thermal resistance can be defined for each section as shown in Figure 1-10.

Heat

source-input Heat sink-_output

Wick Structure Wall Rext.e RPe Rwe Rva Rext.c RPc Rwc RWa Rpa Rint,evap _Rint,cond

12

The total thermal resistance includes all single heat resistances where Rp and Rw represent the

radial conduction in the wall region and wick structure (respectively in evaporator zone and condenser zone). Rext stands for the convection resistance between the external flow and wall,

and Rva, Rpa and Rwa represent the axial conduction through the vapor, wall and liquid in the

wick structure, respectively. Rint,evap and Rint,cond are interfacial resistance between liquid and

vapor related to boiling and condensation. Comparing the orders of the thermal resistances, some of the parallel and serial resistances in the circuit like Rwa, Rpa can be neglected (see

TABLE 1-1). Also, the convection caused by the external flow adds more complexity to the system since a conjugated heat transfer problem need to be solved. By neglecting the wick structure and assuming a constant heat flux at wall, the main thermal resistances in the heat pipe circuit are related to Rva, Rpe, Rpc, Rint.evap and Rint,c [6].

int. int.cond

total evap pe pc va

R =R +R +R +R +R (1.3)

TABLE 1-1: Thermal resistance model in heat pipe and their order of magnitude [6]

Resistance

(K/W) Rw.a→ axial resistance on wick structure 104 Rp.a→ axial resistance of the pipe wall 102 Rv.a→ axial resistance of vapor 10-8 Rwe, Rw.c → radial resistance of liquid in

wick structure (evaporator or condenser) 10+1 Rp.c, Rp.e → radial resistance of the pipe

wall at the evaporator or condenser 10-1 Rint.evap,cond→ liquid-phase interface

radial resistance 10-5

1.6 Heat pipe performance

The performance of the heat pipe is typically estimated from the heat transfer coefficient, thermal resistance and pressure drop. In most published works ([7] and [8]), the thermal resistance is defined as the key characteristic of heat pipe performance. The equivalent thermal resistance is characterized by the temperature difference between the evaporator and condenser divided by the power of evaporator or condenser. The equivalent thermal resistance is calculated with the following equation:

13

(

)

₁

(

)

evap wall cond wall

equiv equiv

evap cond evap cond equiv

T

T

R

R

Po

Po



DL



DL

h

−

=

→

=

=

+

Beside the equivalent thermal resistance, one can define the thermal resistance of the evaporator and condenser, individually.

( )

( )

evap wall sat evap

evap sat cond wall cond cond T T R Po T T R Po − − − = − = (1.5)

1.7 Research project description

This project deals with a heat transfer analysis and optimizing of a heat pipe used in industrial ventilation systems with the help of a numerical model and experimental prototype. The model represents the lab-prototype, which is a smooth U-type pipe half-filled with refrigerant, without any wick structure. The convection on the external surface of the heat pipe is neglected. As mentioned in the previous section, it adds a lot of complexity to the analysis. Additionally, the assumption of assuming a constant heat flux at boundary is acceptable. Therefore, in analysis of heat transfer for a heat pipe, only the inside part is considered.

The heat transfer inside a heat pipe can be divided in different regimes. When the liquid is subcooled, the heat transfer is governed by single-phase forced convection. When heat is increased however, boiling will start in cavities or nucleation sites at the heated walls. The generated bubbles will grow and then detach when a critical size is reached. In the condenser, the saturated vapor in contact with the wall below saturation temperature will condense and thus release its latent heat of vaporization. Comparing to heat conduction in wall and natural convection in the vapor core, the biggest portion of heat transfer is devoted to boiling and condensation. Therefore, an appropriate modeling of the phase changes process is the key step for the study heat transfer in heat pipes.

The first part of this work is the simulation of the bubble nucleation in the evaporation section, which strictly affects the heat transfer rate. There are various works including numerical and experimental that have been done to study boiling phenomena, such as [9] & [10]. The work of these authors became references for the implementation of the boiling model into the commercial software and in OpenFOAM. On the other hand, only a few investigations are

14

published on developing prediction models for condensation phenomena. Therefore, the first object of this PhD thesis is proposing a numerical model compatible with the boiling model (available in OpenFOAM solver) for the simulation of the condensation inside a pipe. After developing an acceptable condensation model, the next step is to simulate a complete heat pipe including an evaporator and condenser working simultaneously. The complexity is important due to the combination of the boiling and condensation models that strongly interact. This challenge becomes even more severe when the system is a closed one, inside which the variation of pressure affect the solution of the equations, drastically. After having validated the numerical model, the thermal resistance is defined as a main characteristic for heat pipe and some sensitivity analysis are performed to seek under which conditions the heat pipe has better performances.

As a part of the project, a heat pipe prototype has been built-up to validate the numerical model with experimental data (Figure 1-11). This physical model is based on a real industrial heat pipe used at Venmar Inc. for large ventilation systems. In this prototype, the working fluid is R-134a and the pipe material is aluminum alloy, the same material as used in the industrial application. This model and its specification will be described in Chapter 5.

Condenser

Evaporator

15

1.8 Objectives

The main purposes of the present PhD project are:

• Develop a new model compatible with the OpenFOAM solver to simulate heat transfer during condensation.

• Combine boiling and condensation models to represent the behaviour of a heat pipe.

• Design, fabricate and perform experimental tests on a setup at UdeS to study the operation of scaled down heat pipe representative of the industrial one.

• Validate the numerical model with experimental data obtained from the tests.

• Investigate the effect of independent parameters on the heat pipe performance and propose an optimized prototype aiming a higher thermal efficiency for the industrial ventilation systems.

• Design two-groove structures for heat pipe and analyze their performance to seek which geometry has better performance.

1.9 Study plan

Although, there are many experimental works to study the heat pipe behaviour, only a few investigations are presented on the development of numerical models. With the help of such models, the influence of important variables such as velocity, pressure, temperature and phase fraction1 _{of each phase can be separately observed. Some models that can be applied for boiling}

and condensation are available in commercial software. Typically, the implemented models take into account the interfacial heat transfer between two phases, in the bulk of the working fluid. However, the wall heat flux, which is one of the most important term, can only be represented with simplified boundary conditions like constant temperature or convection conditions. Thus, such a simplified representation of boiling and condensation at the wall can merely approximate these complex phenomena. Therefore, a numerical approach that generates precise solutions with respect to experimental results is proposed in this work.

16

In the next chapter, the state of the art, the concept of boiling and condensation are reviewed, then, some relations for prediction of heat transfer coefficients for boiling and condensation in different regimes are introduced, which are divided into four main regimes: pool boiling, flow boiling, pool condensation and flow condensation. Finally, the phase change modeling and governing equations are presented.

The numerical solution is obtained through using an open source tool, OpenFOAM, written in C++ and run on Linux systems. The boiling model is already available and validated. However, its concept is comprehensively reviewed in chapter 3 in order to have a better comprehension of phase change model. Some part of this concept is repeated in chapter 4 and 5. Borrowing the idea of boiling model, a new model for simulation of condensation is suggested. The model is introduced in chapter 4 with the related results that have been published in International Journal of Heat and Mass Transfer [11].

After the validation of condensation, both boiling and condensation models are used to simulate the behaviour of a complete heat pipe, as shown in chapter 5. Important difficulties arise when both phenomena work simultaneously in a closed system. The critical point at this stage is the variation of flow parameters like pressure. After having validated the numerical model, some sensitivity analysis on the numerical model are performed to seek the effects of independent parameters on heat pipe function and to propose an optimum prototype. The results of this work have also been published in International Journal of Heat and Mass Transfer [12]. Two designs of grooves inside the heat pipe are proposed and built based on sensitivity analysis conducted with the numerical model and based on literature works. In chapter 6, the improvement of the heat pipe performance in the presence of new groove structure are empirically investigated and the results are compared with the performance of smooth heat pipe described in chapter 5.

Finally, the conclusions of the present work and suggestions for future works are presented in chapter 7.

1.10 Methodology

In order to generate a numerical model representing a two-phase flow in a closed system, a Computational Fluid Dynamics (CFD) methodology is used in this study. The mathematical

17

model is based on conservation equations of Navier-Stokes. PISO2_{, SIMPLE}3 _{and PIMPLE} (combination of PISO and SIMPLE) algorithms are applied in OpenFOAM for the pressure-velocity coupling and to prevent checker-board instability. The solver developed for turbulent flow is selected to calculate turbulent terms generated by chaotic behaviour of bubbles. As shown in TABLE 1-2, there are numerous solvers available for simulation of multiphase flow in OpenFOAM .

TABLE 1-2: Some available solvers in OpenFOAM to model phase changes

Solver name Description Method

InterFoam

Laminar, two incompressible flows, isothermal immiscible fluids

VoF, phase-fraction based interface capturing approach, PISO algorithm

twoPhaseEulerFoam

heat transfer, laminar/turbulent, compressible fluid phases with one phase dispersed, e.g. gas bubbles in liquid

Two-Fluid method, PIMPLE algorithm

compressibleInterFoam compressible, non-isothermal immiscible fluids

using a VoF (volume of fluid) phase-fraction based interface capturing approach

reactingTwoPhaseEulerFoam

system of many compressible fluid, including heat transfer, laminar/turbulent

Two-Fluid method, PIMPLE algorithm, phase change modeling boiling, convection of two phases

The best and most appropriate solver that fits to our problem for modeling the phase change and the interaction between the two phases is the “reactingtwoPhaseEulerFoam”. This solver is enthalpy-based and uses an Euler-Eulerian approach in which each phase is considered as a continuous phase. It means that the boundary conditions are applied for each phase separately and each variable such as velocity, pressure, temperature, and phase fraction is considered for both phases.

The structure of the OpenFOAM code consists of three parts: pre-processing, solver and post-processing (Figure 1-12). In pre-post-processing tool, user need to define the geometry, mesh, fluid

2 _{Pressure Implicit with Splitting of Operator}

18

properties, initial values and boundary conditions. In the solver tool, the governing equations are discretized while in the post-processing tool, results of the solution can be visualized (using tool like Paraview). One of OpenFOAM benefits is that user can add or modify governing equations and extend existing solvers to new needs.

Pre-processing Solver Post-processing

Utilities Meshing tools User Applications Standard Application Paraview Others tools like EnSight Open Source Field Operation and Manipulation (OpenFOAM) C++ library

Figure 1-12: Structure of code in OpenFOAM- [13]

This PhD project is focussed on the optimization of a heat pipe prototype by using a mathematical model validated with experimental data. Since the boiling model is already available in OpenFOAM, there are three main steps to reach final purpose.

• Initial model: First, condensation inside a 3-D channel is separately considered, without the wall region (as shown in Figure 1-13). The nearest existent OpenFOAM solver for modeling phase change is chosen. Wall heat flux and mass generation terms are then added to the general equation. This model is validated by comparing the numerical predictions with experimental data found in open literature.

19

• Second model: After the validation of the condensation model, boiling and condensation models are combined into a complete model of heat pipe, which is a U-type smooth channel with a circular cross section and filling ratio of 50% (as shown in Figure 1-14). The experimental data obtained from the tests is used to validate suggested model in OpenFOAM.

Figure 1-14: Complete model of U-type smooth heat pipe

• Optimized model: After having validated the numerical model, some numerical sensitivity analysis are performed to investigate the effect of independent parameters like size of the tube, applied heat, external mass flow rate of water, type of refrigerant and filling ratio on the heat pipe performance, which can be characterized by its thermal resistance.

20

2 CHAPTER TWO: State of the art

2.1 Phase change concepts

In this section, the concepts of boiling and condensation are briefly reviewed then some correlations for the calculation of heat transfer coefficients are introduced as found in literatures.

2.1.1 Boiling concepts

Boiling happens when at specific pressure, the temperature of a liquid becomes greater than its saturation temperature Tsat and bubbles are then generated. The important parameters in

boiling are the latent heat of vaporization hlv, the surface tension at the liquid-vapor interface

and the thermal properties of the fluid like its thermal conductivity, specific heat, density and viscosity. At the boiling point, the thermal equilibrium is perturbed and bubbles are not in thermodynamic equilibrium with the surrounding liquid. The pressure difference between liquid and vapor is balanced by the surface tension at the interface (Young-Laplace equation). In addition, the temperature difference between newly created bubble and surrounding liquid is the driving force for heat transfer between the two phases. There are two categories of boiling: (a) pool boiling and (b) flow boiling also called forced convection boiling.

2.1.1.1 Pool boiling

Pool boiling is a stationary flow while bubbles grow and rise due to buoyancy effects. Pool boiling consists of four primary regimes: (a) natural convection boiling (no bubbles generated), (b) nucleate boiling (bubbles rise and grow), (c) transition boiling and (d) film boiling [14]. They are briefly explained in the following as presented in Figure 2-1:

• Natural convection boiling (A→B): It looks like evaporation when the liquid temperature is higher than its saturation temperature (∆Tsup=Tl-Tsat is about between 2 oC and 5 oC).

The liquid is superheated and evaporates when it rises to free surface.

• Nucleation boiling (B→D): Bubbles form but collapse and dissipate into the liquid rapidly (subcooled boiling region). The stirring and agitation caused by the entrainment of liquid to the heater surface is responsible for increasing the local heat transfer coefficient. Bubbles can grow and rise to the free surface making numerous continuous columns of vapor (saturated boiling region). At this point, the heat transfer coefficient is tremendously

21

increased by effect of the liquid entrainment and evaporation. The maximum heat transfer coefficient occurs where the heated surface is covered by bubble, (∆T is between 5 o_{C and}

30 o_C).

• Transition boiling (D→E): The heated surface is covered by a vapor film, which acts as a thermal insulator. Due to the low thermal conductivity of vapor relative to liquid, the heat transfer suddenly drops. In reality, boiling cannot follow the transition part of the boiling curve unless applied power is reduced (∆T is between 30 o_{C to 120} o_C).

• Film boiling (E→F): after a drop of the heat transfer coefficient at the end of the transition regime, the radiation of vapor, which is significant at high temperature, causes the heat transfer to rise again (∆T is between 120 o_{C and 1000} o_C).

H ea t F lux (kW /m ² )

2

7

20

100

_T

_-T

A

B

C

E

D

F

convection Nucleation _boiling Transition _regime Film boiling

Subcooled boiling

Saturated boiling

Figure 2-1: Boiling curve, heat flux vs superheat temperature- adapted figure from [15], [9]

The natural convection and nucleation boiling are two main practical regimes that are mostly observed in applications. There are various models in the literature for the prediction of pool boiling in different regimes as mentioned in following table.

22

TABLE 2-1: Proposed correlation for pool boiling regime

Reference Model Regime

Cooper (1984) [16] (0.12 0.2 ) 0.55 0.5 0.67 1.7 55 ( logP ) P , 102.03( 134 ) sf LogC r r wall r sat crt h P M q P P M for R a − − − =  −  = = General Ribatski et Jobardo (2009)[17] 0.3 (0.9 0.3 ) (0.45) 0.8 0.5 0.2 100 ( logP ) Re Pr r r a wall h_{= }P ₋ − M− _q − Nucleation boiling Jung (2003) [18] 1 1.4 0.25 0.309 0.437 1 10( ) (1 T ) Pr 0.0146 2 ( ), 0.855( ) Pr c wall r l l l sat l v r sat crt v l q hD Nu A T g T T T c          − − − = = − = − = = Nucleation boiling Rohsenow (1962)[15] 0.5 3 1.0 ( ) Cp (T ) [ ] [ ] Pr l v v wall sat wall l lv sf lv l g T q h C h     − − = Nucleation boiling Stephan and Abdelsalam (1980) [19] 0.5 0.67 0.248 4.33 2 0.0546[( ) ] [ ] [ ] experimental parameter v wall w lv w l v l l l sat L l L q d h d hD Nu T a a



 









wall wall sat

v wall sat g T q T D T      − + − = − − Film boiling 2.1.1.2 Flow boiling

Flow boiling combines pool boiling with flow convection and is classified into two categories: internal flow and external flow. External flow is similar to pool boiling with additional heat flux due to flow motion. However, in internal flow, vapor and liquid are forced to move together and there is no path for vapor to escape to free surface. Forced convection boiling is commonly referred to as two-phase flow and characterized by rapid changes from liquid to vapor in the flow direction (Figure 2-2).

23

Liquid droplet

Bubbly flow Slug flow Annular flow Mist flow

Bubble in liquid

Transition regime

Liquid flow

vapor liquid

Figure 2-2: Flow boiling regime from stratified to annular regime- adapted figure from [9]

Internal flow regimes can be divided into bubbly, slug, annular, transition and mist flow [20]. In bubbly flow, the bubbles grow and rise from the heated surface. Along the tube, bubble grow in size and eventually coalesce into slug of vapor. Up to half of the volume of the tube can be occupied by the vapor. Later along the tube, an annular flow can be reached and characterized by a liquid that is confined on the walls while the core of the tube is filled with vapor. Such a situation leads to an increase of the heat transfer coefficient. When the annular liquid layer gets thinner, dry spots appear on the surface and decrease the heat transfer because there is no liquid locally; this is transition flow. In the last flow regime, the surface is dried and liquid droplets are suspended in the vapor core until it is vanished; this is the mist flow regime (Figure 2-2). The appearance of dry spots is accompanied by a sharp decrease in the heat transfer coefficient [14]. As described in following table, there are some available correlations in the open literatures for the determination of heat transfer coefficient of flow boiling.

TABLE 2-2: Proposed correlation for flow boiling regime

Reference Model Regime

Shah (1983) [21] 0.5 0.8 0.4 (230 ) 0.023 ( ) Pr , wall sat l sat l l wall l lv T T h h Bo T T q GD h Bo D Gh   − =  + − = = General

24 Wojtan et al. (2006) [22] 0.073 0.24 0.72 2 0.437( v) ( ) wall lo lv l lo L q We Gh D G D We    − − = = General Zhang et al. (2006) [23] 2.31 0.361 0.295 0.311 0.17 0.0352[We 0.0119( ) ( ) ] ( ) [2.05( ) x ] v wall lo l v lv in l L L q D D Gh     − − = + − General Fang(2013) [24] 0.4 0.11 , , 0.95 0.4 2 1.13 .00061( ) Re Pr (Fa) (1.023 ) ( ) , ( ) ( ) 1 30000 , 0.0026 , 36 0.0026 l l l l f l w l v l v wall lv S F hD Nu Ln x Fa F G D x Bo Bo _q S Bo Gh Bo         + = = − = = −  _  =_ =   General Chen (1966)[25] 0.8 0.4 0.79 0.45 0.49 0.24 0.75 0.5 0.29 0.24 0.24 6 1.17 0.736 . . , 0.023Re Pr 0.00122{ } T 1 (1 2.53 10 Re , Re (1 ) 2.35(1 ( 0.213)) 1/ 0.1 1 1/ 0.1 X nb sp sp l l l l l l nb sat sat l lv v l l l tt tt tt tt h S h F h h D Cp h P h S x GD X X F X        − = + = =   = +  = −  _{+ }  =    =₍ v_{) (}0.5 v₎ 0.1₍1 ₎0.9 l l x x     − − Vapor quality between 0.0-0.7 Gungor-Winterton (1987) [26] 2 2 0.86 2 2 0.75 0.41 (0.1 2( ) 2 2 2 2 2 2 2 (0.5) 2 2 2 ( ) , 1 3000( ) 1.12( ) ( ) 1 ( ) ( ) 0.05 1 ( ) ( ) 0.05 1 l wall tp sp lv l v G gD l l l l q h SS FF h S Gh x F x G gD G gD S otherwise G gD G gD F otherwise        − = + = + = −  _  =    _  =   Both vertical and horizontal flows

25 Sun-Mishima (2009)[27] 1.05 0.54 0.191 0.142 2 6 Re ( ) , , Re lo l lo l v wall lo lo l lv l Bo h We D q G D GD We Bo Gh      = = = = General –mini channel Kandlikar (1990)[28] 0.2 0.7 0.8 0.3 0.8 0.5 2 2 0.5 0.667 [0.6683Co f(Fr ) 1058 1.63](1 x) 1 Fr 0.4 f(Fr ) (25 ) Fr 0.4 1 ( ) ( ) , ( / 8)(Re 1000) Pr ( / D) , [0.79 (Re ) 1 12.7(f / 8) (Pr 1) lo lo lo lo lo lo v lo l l lo lo l lo lo lo lo l h Bo h Fr x Co Fr G gD x f h f Ln  _   − = +  −  _  =    − = = − = = + − 2 1.64]− − Both vertical and horizontal flow-general

Since there is no forced flow and thus only natural convection is dominant inside of heat pipe studied in this thesis, a focus is put on pool boiling regimes, especially nucleation boiling.

2.1.2

When the temperature of the vapor is less than Tsat, vapor condenses into liquid. The latent heat

of vaporization, hlv, is released and transferred through the film to the wall surface.

Condensation is divided into two categories: (a) pool condensation, and (b) flow condensation [14].

2.1.2.1 Pool condensation

Pool condensation happens when fluid is stationary and is classified into two regimes: filmwise and dropwise condensation (see Figure 2-3). In filmwise condensation, the liquid droplets wet and cover the whole surface area and create a film layer, which slides down due to the gravity effect. With time, the thickness of this layer which acts a thermal resistance, is increased and the heat transfer rate is thus reduced. Adding as promotor like a chemical coating on the surface, the heat transfer rate can be enhanced and the dropwise regime can be achieved. In dropwise condensation, the small droplets that form at the nucleation sites grow. With time, they coalesce into large droplets, slide down when they reach a certain size and exposing the surface again to vapor. In this case, there is no liquid film that hinders heat transfer. As a result,

26

the heat transfer coefficients in dropwise condensation can become 10 times larger than that associated to filmwise condensation.

5

T

-T

H

ea

t fl

ux

(kW

/m

²

)

Figure 2-3: Pool condensation regime dropwise and filmwise- adapted figure from [29]

In TABLE 2-3, some proposed relations for the calculation of heat transfer coefficients representing pool condensation are presented.

TABLE 2-3: Proposed correlations for film condensation and dropwise regime

Reference Model Regime

Chato (1960) [30] 3 0.25 ( ) 3 0.555[ {h Cp (T T )}] (T T ) D 8 Re 35000 l l v l lv l sat w l sat w v v v v g h U D        − = + − − =  Filmwise condensation-inside horizontal tube Nusselt (1960) [31] 3 0.25 ( ) 3 0.729[ {h Cp (T T )}] (T T ) D 8 Re 1800 l l v l lv l sat w l sat w l l l l g h U D        − = + − − =  Filmwise condensation-outside horizontal tube

27 Cooper P. Griffith (1965) [31] 51104 2044 22 100 255310 100 o o sat sat o sat T C T C h T C  _{+  }  =    Dropwise condensation steam on copper surface 2.1.2.2 Flow condensation

For flow condensation inside a horizontal tube, gravity and inertial forces are the two primary driving forces influencing the flow regime, which can be classified into four categories: stratified, stratified-wavy, transient and annular regime. In stratified and stratified-wavy flow regime, the gravitational force is more dominant than the inertial force, causes liquid film accumulates at the bottom of tube. The liquid film then acts as a thermal resistance and decreases the heat transfer coefficient. At higher velocities as the flow is entering annular regime, the shear stress forces control the flow, the liquid film thickness is diminished, and the heat transfer coefficient is increased [32]. Other types of flow regime, like slug or mist flow, can be possible but are less frequent than stratified and annular regimes (see Figure 2-4).

vapor flow A vapor liquid A B B C C B-B Stratified A-A Annular C-C Bubbly vapor flow Plug Slug Wavy annular Mist annular

Figure 2-4: Flow condensation regime- adapted figure from [29]

Some suggested correlations for different flow regime of flow condensation is presented in TABLE 2-4.